Let A, B, and C be any sets with A ⊆ C and B ⊆ C. Show that A – B = A ∩ (C – B). Can you help me prove the proof holds true?

I assume A - B is the complement of B in A, i.e. all elements of A that do not belong to B.

To prove equality between two sets, you have to show they are subsets of one another. To prove one set is a subset of another set, you have to show that any element in the first set belongs to the second set.

So, to show that A - B = A ∩ (C - B), you have to prove

• A - B ⊆ A ∩ (C - B)

• A ∩ (C - B) ⊆ A - B

Left to right:

Let x ∈ A - B.

By definition of set complement, x ∈ A and x ∉ B.

Since A ⊆ C,  x ∈ C.

So x ∈ C - B, and by definition of set intersection, x ∈ A ∩ (C - B).

This proves A - B ⊆ A ∩ (C - B).

Right to left:

Let x ∈ A ∩ (C - B).

By definition of set intersection, x ∈ A and x ∈ C - B.

By definition of set complement, x ∈ C and x ∉ B.

Again by definition of complement, x ∈ A and x ∉ B means that x ∈ A - B.

This proves A ∩ (C - B) ⊆ A - B.

Both sets are subsets of one another, so A - B = A ∩ (C - B).